Problem 49 - Precalculus Essentials

Question

Express the exact value of each function as a single fraction. Do not use a calculator. $$\text { If } f(\theta)=2 \cos \theta-\cos 2 \theta, \text { find } f\left(\frac{\pi}{6}\right)$$

Step-by-Step Solution

Answer

The answer is \(f(\pi/6) = \sqrt{3}/2\).

Step 1: Apply the double-angle identity

The given function is \( f(\theta) = 2 \cos \theta - \cos 2 \theta \). Express \( \cos 2 \theta \) in terms of \( \cos \theta \) by using the identity \( \cos 2 \theta = 2 \cos^2 \theta - 1 \). Hence the function becomes \( f(\theta) = 2 \cos \theta - (2 \cos^2 \theta - 1) \). This simplifies to \( f(\theta) = 2 \cos \theta - 2 \cos^2 \theta + 1 \).

Step 2: Substitute the given theta

Now substitute \( \theta = \pi/6 \) into the function. It becomes \( f(\pi/6) = 2 \cos (\pi/6) - 2 \cos^2 (\pi/6) + 1 \).

Step 3: Simplify the function

By knowing that \( \cos (\pi/6) = \sqrt{3} / 2 \), plug this into the equation. This gives \( f(\pi/6) = 2 (\sqrt{3} / 2) - 2 (\sqrt{3} / 2)^2 + 1 \). Simplify it and get the answer.